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Theorem coseq00topi 20400
Description: Location of the zeroes of cosine in  ( 0 [,] pi ). (Contributed by David Moews, 28-Feb-2017.)
Assertion
Ref Expression
coseq00topi  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )

Proof of Theorem coseq00topi
StepHypRef Expression
1 simpl 444 . . . . . 6  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  ( 0 [,] pi ) )
2 0re 9081 . . . . . . 7  |-  0  e.  RR
3 pire 20362 . . . . . . 7  |-  pi  e.  RR
42, 3elicc2i 10966 . . . . . 6  |-  ( A  e.  ( 0 [,] pi )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
51, 4sylib 189 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
65simp1d 969 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  e.  RR )
73a1i 11 . . . . 5  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  pi  e.  RR )
87rehalfcld 10204 . . . 4  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( pi  /  2
)  e.  RR )
96, 8lttri4d 9204 . . 3  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  (
pi  /  2 )  \/  A  =  ( pi  /  2 )  \/  ( pi  / 
2 )  <  A
) )
10 simplr 732 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =  0 )
116ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  RR )
12 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  A )
13 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  <  ( pi  /  2
) )
142rexri 9127 . . . . . . . . . . 11  |-  0  e.  RR*
15 halfpire 20365 . . . . . . . . . . . 12  |-  ( pi 
/  2 )  e.  RR
1615rexri 9127 . . . . . . . . . . 11  |-  ( pi 
/  2 )  e. 
RR*
17 elioo2 10947 . . . . . . . . . . 11  |-  ( ( 0  e.  RR*  /\  (
pi  /  2 )  e.  RR* )  ->  ( A  e.  ( 0 (,) ( pi  / 
2 ) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) ) )
1814, 16, 17mp2an 654 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <  ( pi  /  2 ) ) )
1911, 12, 13, 18syl3anbrc 1138 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  A  e.  ( 0 (,) (
pi  /  2 ) ) )
20 sincosq1sgn 20396 . . . . . . . . 9  |-  ( A  e.  ( 0 (,) ( pi  /  2
) )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2119, 20syl 16 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  (
0  <  ( sin `  A )  /\  0  <  ( cos `  A
) ) )
2221simprd 450 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  0  <  ( cos `  A
) )
2322gt0ne0d 9581 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  < 
A )  ->  ( cos `  A )  =/=  0 )
24 cos0 12741 . . . . . . . 8  |-  ( cos `  0 )  =  1
25 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  0  =  A )
2625fveq2d 5724 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  0 )  =  ( cos `  A
) )
2724, 26syl5reqr 2482 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =  1 )
28 ax-1ne0 9049 . . . . . . . 8  |-  1  =/=  0
2928a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  1  =/=  0 )
3027, 29eqnetrd 2616 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  /\  0  =  A )  ->  ( cos `  A )  =/=  0 )
315simp2d 970 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  <_  A )
322a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
0  e.  RR )
3332, 6leloed 9206 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
3431, 33mpbid 202 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( 0  <  A  \/  0  =  A
) )
3534adantr 452 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( 0  <  A  \/  0  =  A ) )
3623, 30, 35mpjaodan 762 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  ( cos `  A )  =/=  0
)
3710, 36pm2.21ddne 2672 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  <  (
pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
38 simpr 448 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  A  =  ( pi  /  2 ) )  ->  A  =  ( pi  /  2
) )
39 simplr 732 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =  0 )
406ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  RR )
41 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( pi  /  2 )  <  A
)
42 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  <  pi )
433rexri 9127 . . . . . . . . . . 11  |-  pi  e.  RR*
44 elioo2 10947 . . . . . . . . . . 11  |-  ( ( ( pi  /  2
)  e.  RR*  /\  pi  e.  RR* )  ->  ( A  e.  ( (
pi  /  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) ) )
4516, 43, 44mp2an 654 . . . . . . . . . 10  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  <->  ( A  e.  RR  /\  ( pi 
/  2 )  < 
A  /\  A  <  pi ) )
4640, 41, 42, 45syl3anbrc 1138 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  A  e.  ( ( pi  / 
2 ) (,) pi ) )
47 sincosq2sgn 20397 . . . . . . . . 9  |-  ( A  e.  ( ( pi 
/  2 ) (,) pi )  ->  (
0  <  ( sin `  A )  /\  ( cos `  A )  <  0 ) )
4846, 47syl 16 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( 0  <  ( sin `  A
)  /\  ( cos `  A )  <  0
) )
4948simprd 450 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  <  0
)
5049lt0ne0d 9582 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  <  pi )  ->  ( cos `  A )  =/=  0
)
51 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  A  =  pi )
5251fveq2d 5724 . . . . . . . 8  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  ( cos `  pi ) )
53 cospi 20370 . . . . . . . 8  |-  ( cos `  pi )  =  -u
1
5452, 53syl6eq 2483 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =  -u
1 )
55 ax-1cn 9038 . . . . . . . . 9  |-  1  e.  CC
5655, 28negne0i 9365 . . . . . . . 8  |-  -u 1  =/=  0
5756a1i 11 . . . . . . 7  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  -u 1  =/=  0 )
5854, 57eqnetrd 2616 . . . . . 6  |-  ( ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  /\  A  =  pi )  ->  ( cos `  A )  =/=  0
)
595simp3d 971 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  <_  pi )
606, 7leloed 9206 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <_  pi  <->  ( A  <  pi  \/  A  =  pi )
) )
6159, 60mpbid 202 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  -> 
( A  <  pi  \/  A  =  pi ) )
6261adantr 452 . . . . . 6  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( A  <  pi  \/  A  =  pi ) )
6350, 58, 62mpjaodan 762 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  ( cos `  A )  =/=  0
)
6439, 63pm2.21ddne 2672 . . . 4  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( pi  / 
2 )  <  A
)  ->  A  =  ( pi  /  2
) )
6537, 38, 643jaodan 1250 . . 3  |-  ( ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A
)  =  0 )  /\  ( A  < 
( pi  /  2
)  \/  A  =  ( pi  /  2
)  \/  ( pi 
/  2 )  < 
A ) )  ->  A  =  ( pi  /  2 ) )
669, 65mpdan 650 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  ( cos `  A )  =  0 )  ->  A  =  ( pi  /  2 ) )
67 fveq2 5720 . . . 4  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  ( cos `  (
pi  /  2 ) ) )
68 coshalfpi 20367 . . . 4  |-  ( cos `  ( pi  /  2
) )  =  0
6967, 68syl6eq 2483 . . 3  |-  ( A  =  ( pi  / 
2 )  ->  ( cos `  A )  =  0 )
7069adantl 453 . 2  |-  ( ( A  e.  ( 0 [,] pi )  /\  A  =  ( pi  /  2 ) )  -> 
( cos `  A
)  =  0 )
7166, 70impbida 806 1  |-  ( A  e.  ( 0 [,] pi )  ->  (
( cos `  A
)  =  0  <->  A  =  ( pi  / 
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   RRcr 8979   0cc0 8980   1c1 8981   RR*cxr 9109    < clt 9110    <_ cle 9111   -ucneg 9282    / cdiv 9667   2c2 10039   (,)cioo 10906   [,]cicc 10909   sincsin 12656   cosccos 12657   picpi 12659
This theorem is referenced by:  coseq0negpitopi  20401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7469  df-card 7816  df-cda 8038  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-ioc 10911  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-fl 11192  df-seq 11314  df-exp 11373  df-fac 11557  df-bc 11584  df-hash 11609  df-shft 11872  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-limsup 12255  df-clim 12272  df-rlim 12273  df-sum 12470  df-ef 12660  df-sin 12662  df-cos 12663  df-pi 12665  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-sca 13535  df-vsca 13536  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-hom 13543  df-cco 13544  df-rest 13640  df-topn 13641  df-topgen 13657  df-pt 13658  df-prds 13661  df-xrs 13716  df-0g 13717  df-gsum 13718  df-qtop 13723  df-imas 13724  df-xps 13726  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-submnd 14729  df-mulg 14805  df-cntz 15106  df-cmn 15404  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-fbas 16689  df-fg 16690  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cld 17073  df-ntr 17074  df-cls 17075  df-nei 17152  df-lp 17190  df-perf 17191  df-cn 17281  df-cnp 17282  df-haus 17369  df-tx 17584  df-hmeo 17777  df-fil 17868  df-fm 17960  df-flim 17961  df-flf 17962  df-xms 18340  df-ms 18341  df-tms 18342  df-cncf 18898  df-limc 19743  df-dv 19744
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